## Characteristics and the three gap theorem.(English)Zbl 0709.11011

Let [x] and $$\{$$ $$x\}$$ denote the integer and fractional parts of the positive real number x. If $$d_ m=[(m+1)x]-[mx]$$ then $$[mx]=\sum^{m- 1}_{i=1}d_ i+[x]$$. Since each $$d_ i$$ is [x] or $$[x]+1$$, we can define the characteristic of x to be the string $$x_ 1x_ 2x_ 3..$$. where $$x_ i=s$$ if $$d_ i=[x]$$ and $$x_ i=\ell$$ if $$d_ i=[x]+1$$ (s for small and $$\ell$$ for large). The authors exhibit a connection, for some x, between the characteristic of x and the sequence of arcs or gaps formed by partitioning a circle by successive placements of points by an angle of x revolutions. They do this by making use of the three gap theorem, which asserts that points placed in this way partition a circle into gaps of 2 or 3 different lengths.
Reviewer: Ian Anderson

### MSC:

 11B05 Density, gaps, topology 11A55 Continued fractions

### Keywords:

greatest integer; fractional parts; three gap theorem